3. Subalgebras of Algebras

We have already noted that we can recognize \(A_2 = su(3)\) as a subalgebra of \(A_3 = su(4)\), \(B_3 = so(7)\), and \(C_3 = sp(2\cdot 3)\) by identifying its hexagonal root diagram in each of the larger root diagrams. This section develops two further methods which help us identify subalgebras using root and weight diagrams. We emphasize that these methods are valid only when applied to the full root diagram, but not when applied to the root system alone.

3.1 Subalgebras of \(B_3=so(7)\) using Root Diagrams

Root and weight diagrams can be used to identify subalgebras. For algebras of rank \(l \le 3\), we can recognize subdiagrams corresponding to subalgebras by building the algebra's root and weight diagrams and rotating them in \(\mathbb{R}^3\).

We illustrate the process of finding all subalgebras of \(B_3=so(7)\) by using its root diagram. When rotated to the position in Figure 11, the large horizontal 2-dimensional rectangle through the origin (containing roots colored green and magenta) contains eight non-zero nodes. This subdiagram is the root diagram of the 10-dimensional algebra \(B_2 = C_2\). The smaller rectangular diagrams lying in parallel planes above and below this large rectangle contain all of the roots vectors of \(B_2=C_2\). These diagrams are minimal weight diagrams of \(B_2 = C_2\).

Two additional rotations of the root diagram of \(B_3=so(7)\) produce subalgebras. In Figure 12, the horizontal plane through the origin contains only two orthogonal roots, which are colored blue and fuchsia. These roots comprise the root diagram of \(D_2=su(2)+su(2)\). We have already identified the hexagonal \(A_2=su(3)\) root diagram in the horizontal plane containing the origin in Figure 13. Each horizontal triangle above and below this plane is a minimal weight representation of \(A_2\). In addition, the subdiagram containing the bottom triangle, middle hexagon, and top triangle is the 15-dimensional, rank 3 algebra \(A_3 = D_3\). Thus, it is possible to identify both rank \(l-1\) and rank \(l\) subalgebras of a rank \(l\) algebra.

Figure 11. \(B_2 \subset B_3\)

Figure 12. \(D_2 \subset B_3\)

Figure 13. \(A_2 \subset D_3 \subset B_3\)

Identifying Subalgebras of \(B_3=so(7)\) using Subdiagrams of its Root Diagram

Not all subalgebras of a rank \(l\) algebra can be identified as subdiagrams of its root or weight diagram. When extending a rank \(l\) algebra to a rank \(l+1\) algebra, each root \(r^i = \langle \lambda_1^i, \cdots, \lambda_l^i \rangle\) is extended in \(\mathbb{R}^{l+1}\) to the roots \(r^{i_1}, \cdots, r^{i_m}\), where \(r^{i_j} = \langle \lambda_1^i, \cdots, \lambda_l^i, \lambda_{l+1}^{i_j} \rangle\). Here, \(\lambda_{l+1}^{i_j}\) is one of \(m \ge 1\) different eigenvalues values defined by the extension of the algebra. Hence, although roots \(r^{i_1}, \cdots, r^{i_m}\) are all distinct in \(\mathbb{R}^{l+1}\), they are the same root when restricted to their first \(l\) coordinates. Thus, we can identify subalgebras of a rank \(l+1\) algebra by projecting its root and weight diagrams along any direction.

Just as our eyes ``see'' objects in \(\mathbb{R}^3\) by projecting them into \(\mathbb{R}^2\), we can identify subalgebras of \(B_3=so(7)\) by projecting its root and weight diagram into \(\mathbb{R}^2\). In Figure 14 and Figure 15, we have rotated the root diagram of \(B_3\) so that our eyes project one of the root vectors onto another root vector. This method allows us to identify \(B_2=so(5)\) and \(G_2\) as subalgebras of \(B_3\) by recognizing their root diagrams in Figure 14 and Figure 15, respectively. Although we previously identified \(B_2 \subset B_3\) by using a subdiagram confined to a plane, we could not identify \(G_2\) as a subalgebra of \(B_3\) using that method.

Figure 14. \(B_2=so(5) \subset B_3=so(7)\)

Figure 15. \(G_2 \subset B_3=so(7)\)

Identify Subalgebras of \(B_3=so(7)\) using Projections of its Root Diagram